Computer circuits for processing trigonometric data



Feb. 3, 1970 J, HEAIVISIDE ETAL 3,493,735

COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA Filed March 20, 1964 15 Sheets-Sheet 1 SLO i IO 030- CONViRTER 2 0rc7ok k; 0 i

5x7: BRID6 ca/vnzaz. ADJUST ANGLE REAP our Fla-i R SIMULATOR CONVERTER 3 *Cfi R5, FORM "'C8 52 l. 3 IL LLLQJLQI 6 a INVENTORS JOHN B. HEAVISIDE FRANKLIN W. SMITH Jr. BY

MORGAN, FINNEGAN DURHAM 8: PINE ATTORNEYS 1 3,493,735 COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIO DATA Filed March 20, 1964 1.970 B HEAVISIDE F-TAL eet 2 1,5 Sheets-Sh INVENTORS JOHN B. HEAVISIDE FRANKLIN W. SMITH Jr.

MORGAN, FINNEGAN, DURHAM 8| PINE ATTORNEYS COMPUTER CIRCUITS FOR PROCESSING TRIIGONOMETRIC DATA Filed March 20, 1964 1970 J. B. HEAVlSlDE ETAL 1 5 Sheets-Sheet 5 MN QE Q MQ INVENTORS JOHN B. HEAVISIDE FRANKLIN W. SMITH Jr.

MORGAN, FINNEGAN, DURHAM 6 PINE ATTORNEYS bv 1970 J. B.HEAVISIDE ETAL 3,493,735

COMPUTER CIRCUITS FOR PROCESSING TRIGONOME TRIC DATA Filed March 20, 1964 1,5 Sheets-Sheet 4 INVENTORS JOHN B. HEAVISIDE FRANKLIN W. SMITH Jr.

BY MORGAN, FINNEGAN, DURHAM 8| PINE ATTORNEYS Feb. 3, ,1970 J B HEAV|$|DE ETAL 3,493,735

COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA Filed March 20, 1964 1.5 Sheets-Sheet 5 mm 0.. M W q w+C -e m k u k N Qm rm mx C T m m J I Yw hv? m AYqweShW. My

INVENTOR5 JOHN B.HEAV|S|DE BY FRANKLIN w. 5mm Jr. MORGAN, FINNEGAN, DURHAM 8| PINE ATTORNEYS 15 Sheets-Sheet 6 Qm QE Feb. 3, 1970 .1. B. HEAVISIDE A COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA Filed March 20. 1964 mommu INVENTORS JOHN B. HEAVISlDE FRANKLIN w. SMITH Jr. BY 1 MORGAN, FINNEGAN, DURHAM 8 PINE ATTORNEYS Feb. 3, 1970 J. B. HEAVISIDE ETAL 3,

COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA l5 Sheets-Sheet '7 ATTQRNEYS 1.5 Sheets-Sheet 8 J. B. HEAVISIDE ETAL INVENTORS JOHN B. HEAVISIDE FRANKLIN w. smm Jr. BY

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Feb. 3, 1970 COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA Filed March 20, 1964 J. B. HEAVISIDE L 3,493,735

Feb. 3, 1970 COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA 1,5 Sheets-Sheet 9 Filed March 20, 1964 TENTHS OF MINUTES OR TENS OF SECONDS I MINUTES I I TENTHS OF MINUTES FIG.'3F

INVENTORS JOHN B. HEAVISIDE FRANKLIN W. SMITH Jr.

MORGAN, FINNEGAN,DURHAM 8| PINE ATTORNEYS Feb. 3, 1970 ,1. B. HEAVISIDE ETAL 3,493,735

COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA Filed March 20, 1964 l5 Sheets-Sheet l0 BY MORGAN, FINNEGAN, DURHAM 8 PINE ATTORNEYS Feb. 3, 1970 J. B. HEAVISIDE ETAL 3,493,735

COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA Filed arch 20, 1964 5 Sheets-Sheet 11 6;? f 7 '0 Q I x E u A A f L D- Q I I 2- I Q I 9 K 10 Q n 2 2 I 3] a 2 5-5 IL s Q N INVENTORS JOHN B. HEAVISIDE FRANKLIN W. SMITH Jr. BY

MORGAN, FINNEGAN, DURHAM QPINE ATTORNEYS Feb. 3, 1970 J. B. HEAVISIDE 3,493,735

COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA Filed March 20, 1964 l5 Sheets-Sheet 13 9 Mm M m C m a 00 m a a 55 c c MN '0 I68 m y All, 5 a MA a HIGH RA 7/0:

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. fin M 7 mm a G F u s MR ML 7 EW 0 s a m C RE BL 1 I, m W 5 6 E T INVENTORS JOHN B. HEAVISIDE FRANKLIN w. sm'rn Jr. 1 BY MORGAN, FINNEGAN, DURHAM 8 PINE ATTORNEYS I b m M m c 0 d j \I M xk kukwmviou $66K I l I Feb. 3, 1970 J. B. HEAVISIDE ETAL 3,493,735

COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA Filed March 20, 1964 l5 Sheets-Sheet 14 TO ERROR DETECTOR BR/D' INVENTORS ADJUST JOHN B. HEAVISIDE E... 3 FRANKLIN w. SMITH Jr.

MORGAN, FINNEGAN, DURHAM a PINE ATTORNEYS Feb. 3, 1970 J. B. HE kVISIDE AL 3,

COMPUTER CIRCUITS FOR PROCESSING TRIGONOMETRIC DATA Filed March 20, 1964 r l5 Sheets-Sheet 15 FRQM ,amoee ADJUST I I 1 a a Q f i I a J! I x g 2 i i u.

INVENTORS JOHN B. HEAVISIDE FRANKLIN W. SMITH Jr. BY

MORGAN, FINNEGAN, DURHAM 8| PINE ATTORNEYS United States Patent US. Cl. 235179 64 Claims ABSTRACT OF THE DISCLOSURE Computer circuits for processing trigonometric data are disclosed herein and include synchro/resolver converters, bridges and simulators; static transformer circuits for selectively converting synchro data to resolver data and vice versa are described together with manual and automatic bridge arrangements. Special nondecimal dividers and a to d and d to a conversion are also disclosed.

This invention relates to techniquies for the processing of data and is particularly but not exclusively concerned with techniques for the measurement, generation and other processing of data related to functions of angles including trigonometric functions.

The invention is further related to novel components and techniques, and combinations thereof, which in the illustrated embodiments are applied to the conversion of synchro data to resolver data and vice versa, to the formation of ratios, to the measurement of trigonometric data, to function generation, to digital/ analog and analog digital translation, and to combinations of these and related functions.

It is an object of the invention to provide methods and means for effecting these functions with both precision and economy. Further objects are to provide versatility whereby data in a variety of forms may be processed with a minimum number of components, and to provide a high order of accuracy, e.g., accuracy to one part per million, without :a concomitant increase in complexity.

Other objects and advantages of the invention will be set forth in part hereinafter and in part will be obvious herefrom, or may be learned by practice with the invention, the same being realized and attained by means of the instrumentalities and combinations pointed out in the appended claims.

Serving to illustrate exemplary embodiments of the invention are the drawings of which:

FIGURE 1 is a schematic block diagram illustrating a bridge configuration for the measurement of angular data;

FIGURES 2A through 2E inclusive are schematic diagrams of form converters for converting synchro type data to resolver data and vice versa;

FIGURES 3A and 3B are schematic diagrams illustrating resolver bridge arrangements;

FIGURE 30 is a plot of certain error functions related to bridge and simulator data processing;

FIGURE 3D is a schematic diagram of a resolver type bridge energized via a form converter and isolator;

FIGURE 3E is a schematic diagram of a synchro type bridge energized via a form converter and isolator;

FIGURE 3F is a schematic diagram of a divider arrangement for yielding data in the dimensions of minutes and seconds, or minutes and fractions of minutes;

FIGURE 4 is a schematic block diagram of a simulator arragement for generating signals as a function of angular input data;

"ice

FIGURE 5 is a schematic diagram illustrating in greater detail a simulator and converter/isolator;

FIGURE 6 is a schematic diagram of a simulator employing a synchro type simulator and a converter/ isolator; and

FIGURES 7 and 7A7C are schematic diagrams of automatic bridge systems and components including digital switching arrangements therein.

GENERAL The data processing systems according to the invention include means for computing angular data from signals representing vector quantities. Typically, such signals are of analog form and are provided by synchro or resolver equipment; the data are thus represented in synchro or resolver form.

A bridge system for computing the angle represented by a set of synchro or resolver voltages is shown in FIG- URE 1. Resolver voltages are applied across terminals RS RS and RS RS of a form converter/isolator 1. These complementary voltages may be characterized by the general form E sin 0 and E cos 0, respectively. Alternatively, synchro voltages may be applied to the converter across terminals S S and S These complementary voltages, as applied across the terminal pairs, may be characterized by the general form E sin 6, E sin [6+l20], and E sin [0+240].

As described more fully hereinafter the form converter is provided with switching means for selecting a mode of operation which depends upon whether the input constitutes synchro voltages or resolver voltages. The output of the form converter is supplied to a bridge circuit 2 which may be illustratively a resolver bridge or synchro bridge. In those arrangements where the bridge is of the resolver type then the form converter 1 is operative to convert synchro voltages into equivalent resolver type voltages for application to the bridge. Isolation of the bridge from the input system is concurrently provided.

With resolver type voltages applied to the form converter of the above example, the converter is switched in such a manner as to provide isolation alone since the conversion function is not required. In this manner both synchro and resolver data may be processed, with isolation jointly provided.

In the event that the bridge 2 is of the synchro type then the form converter is of a configuration which converts resolver data to equivalent synchro data so that the bridge may compute the angle of the input for both resolver and synchro inputs. Hence, irrespective of the type of bridge involved, when used with the appropriate form converter, the combination is capable of processing both synchro and resolver data.

The bridge circuit 2 is adjusted by an appropriate control system 3 which may be manual or operated by suitable signals from an external point. For example, the bridge may be adjusted via a program system e.g., to receive command angle data as symbolized by Ext. control. In the case of manual adjustment a null detector 4 which is fed from the bridge output is observed and when the null condition is achieved the bridge adjustment is read on indicator 5. The angle thus indicated represents the value of the input data angle 0 represented in either the synchro or resolver inputs. Although the computed and displayed angle is illustrated in degrees and decimal fractions thereof, provision may be made as described more fully herein-after for computing and displaying the angle in terms of degrees, minutes and seconds or fractions of minutes or in any other angular dimensional units.

For automatic operation, the bridge adjustment system 3 includes a control arrangement responsive to the output of the error detector 4 and adapted to automatically adjust the bridge to achieve a null, e.g., by a suitable scanning and switching combination. The bridge switching circuits may be of the type involving binary coded configurations in which the turns ratios of inductive dividers are varied by suitable switching.

Other functions according to the invention are concerned with generating angle-representing or vector data which illustratively are in resolver or synchro form. Since this function generation is analogous to the operation of synchro and resolver equipment, some components of the arrangements are sometimes referred to as simulators. The term standards is also frequently used. A system involving these functions is shown in FIGURE 4.

In the arrangement of FIGURE 4 a simulator 6 is adjusted manually or remotely by an adjustment arrangement 7 which sets the simulator at the desired angle. The angle may be displayed on read-out indicator 8. The displayed angle may be as shown or may have other dimensions as noted in connection with FIGURE 1.

The simulator generates voltages in accordance with the angle to which it is set and these voltages are supplied to form converter 1. The converter also provides isolation and may be switched, as noted in connection with FIGURE 1, between an isolating mode and an isolating/ converter mode, depending upon the type of simulator employed. Where the simulator is of the synchro type then the converter is of the type which converts synchro data to resolver data. Hence, either resolver or synchro functions may be generated. Similarly, where the simulator is of the resolver type then the converter is configured to provide resolver-to-synchro conversion so that with mode switching, both synchro and resolver function generation is attained. Resolver voltages are developed at output terminals RS RS and RS RS the synchro voltages appear at terminals S S and S FORM CONVERTERS Exemplary form converters are shown in FIGURES 2A through 2E. All of the illustrated embodiments have a bilateral characteristic, i.e., they are capable of converting sycnhro type data to resolver type data and vice versa. They are also adapted for switching as more specifically described hereinafter to provide isolating functions alternatively to conversion and isolation.

Referring to FIGURE 2A, the application of synchro data [E sin 0, E sin (+l20"), and E sin (6+240)] at synchro terminals S S and S results in the appearance of the equivalent resolver data at resolver terminals RS RS [Where E sin 0 appears], and RS RS [where E cos 0 appears].

The converse operation transforming resolver inputs to synchro outputs is also available as indicated by the double headed arrow.

The converter of FIGURE 2A comprises three transformers T T and T having windings W W and W respectively, on the synchro side; these windings are interconnected in delta configuration with synchro terminals S S and S Transformers T T and T also have respective resolver windings W W and W On the synchro side, terminal S is connected to V and W S is connected to W and W and S is connected to W, and W The voltage induced in W of T from W yields resolver voltage E sin 0 at terminals RS and RS Resolver windings W and W are connected in series- Opposition to develop the cosine function, E cos 0; this voltage appears across terminals RS and RS When the arrangement of FIGURE 2A is operated either to convert synchro data to resolver data or vice versa, the turns ratios are as follows:

K may have any desired value, and windings may be adjusted or tapped as required for the isolation of either synchro or resolver data. For example, W may be tapped at 10 to yield three identical transformation ratios for synchro isolation.

From the foregoing and With particular values of K assigned, the relationship between the argument E of the synchro data and E of the resolver data may be evaluated. The foregoing ratios can be shown to be the result of projecting Cartesian coordinate data characterizing resolver voltages on the synchro axis which are separated by 120 intervals, and by the reverse operation.

It may be noted that the arrangement of FIGURE 2A includes two identical transformers T and T These transformers may thus serve effectively as isolation transformers. As will be shown, the resultant isolation function is particularly useful in connection with the resolver bridges described hereinafter.

In addition to the foregoing, the circuit of FIGURE 2A may be implemented with three identical synchro windings facilitating balanced impedance conditions. Moreover, the turns ratios are not substantially different thus facilitating overall symmetry in impedance conditions. The delta configuration of the synchro windings when contrasted with a Y connection provides a measure of independence from the effects of adverse voltage division due to lack of exact similarity among the three synchro windings. For this reason delta connections are particularly suitable in bridge applications.

To facilitate the isolation function in certain applications, windings W and W may be provided with additional sections W and W as seen in FIGURE 2B. As noted more fully hereinafter these additional winding sections may be utilized for providing isolation when the converter is switched out of the conversion mode. For typical applications the turns ratios in the arrangement of FIGURE 2B are as follows:

The above ratios are applicable for either synchro-to resolver conversion or the converse.

FIGURE 2C illustrates the arrangement of FIGURE 2A [with the synchro and resolver windings shown in interchanged locations] and wherein the synchro windings W and W are provided with taps at 2' and 3, re spectively, such that sections W and W have turns ratios with respect to W and W of 11K. This arrangement facilitates the switching between form conversion and isolation in certain applications. Apart from the foregoing the circuit of FIGURE 2C is identical to the circuits of FIGURES 2A and 2B.

The converter of FIGURE 2D comprises three transformers T T and T having Y connected synchro windings W W and W respectively. These windings are connected to synchro terminals S S and S The transformers T and T include resolver windings W W and W W respectively; transformer T has a single resolver winding W Resolver windings W and W are connected in seriesopposttlon to terminals RS and RS for providing the resolver sine function, E, sin 0. windings W and W are connected in series-aiding and the combination is connected in series-opposition to W This overall combmation is connected to terminals RS and RS to yield E cos 0. The ratios for synchro-to-resolver conversion are as follows:

An analysis of the arrangement of FIGURE 2D shows that it provides balanced impedance conditions with respect to the synchro terminals. The Y connection yieds relatively high input impedance and transformers T and T being identical, are especially adapted to provide the isolation function. Since each of these transformers includes two resolver windings, a choice of transfer ratios is available.

The converter of FIGURE 2E comprises a pair of transformers T and T having windings interconnected in a manner which resembles the so-called Scott connection which is utilized for converting a time phase system of one number to a time phase system of a different number of phases. The synchro side of T comprises a winding W which is tapped to provide sections WI and Wqb. Transformer T is provided with a single synchro winding W which is connected at one end to synchro terminal S and at its other end to the junction of W and W The outside terminals of W are connected to synchro terminals S and S so that the synchro side is Y connected.

The transformers include respective resolver windings W and W Winding W- of T is connected to resolver terminals RS and RS for supplying the voltage E sin 6. W provides E cos at terminals RS; and RS Suitable turns ratios for synchro-to-resolver conversion are as follows:

Balanced impedance and isolation functions are attainable in the arrangement of FIGURE 2E by proper choice of the value of K and by providing an accessible tap on W so that T and one primary-secondary transformer combination of T (e.g., Wm, W are equal.

It may be seen from the foregoing that by virtue of the independent adjustability of the individual transformers, the converters may provide either balanced input or output impedances as required by their application in bridges and simulators.

In addition to the foregoing converters, the so-called Woodbridge type connection utilized in time domain polyphase converters may be employed for the mechanical phase conversion herein described. Additional circuits are also shown in connection with specific bridge and simulator arrangements hereinafter described.

BRIDGES AND BRIDGE-CONVERTER SYSTEMS Resolver bridges are adapted to compute the angle 0 from inputs e e of the form:

e '=R sin 0 1) In the general case R is also unknown and includes the vector amplitude E and appropriate scaling and dimension coefficients. Both relationships (B and (B are therefore required in order to derive 0.

In a known prior art divider technique the ratio e /e is measured with a linear decade inductive divider, say of five or six decades, whereby the divider reading representing tangent 0 or cotangent 6 is derived. This is accomplished by applying one of the voltages, e.g., e across the entire divider; a null device has one terminal connected to e; and the other terminal connected to the movable contact of the divider. When the movable contact is adjusted through a range of trial ratios, to obtain a null, the divider is set at the ratio 8 /6 which represents tan 0.

6 Of course, interchange of 6 and e will be necessary when the ratio e /e is outside the range of the divider. Polarities must also be controlled.

Obtaining tangent 0 is only a part of the described solution. The ratio, e.g., 0.57735 must be translated into the actual angle, e.g., 30 by consulting a table of tangent or cotangent functions. This mental or manual step places a serious limitation on the described technique since it rules out practice use in automatic processing systems and further, because it creates the hazard of error and runs counter to the arts preference for self-sufiicient instruments. While there are known mechanical and electrical techniques for the machine conversion of tangent 0 to 0 so that the divider setting can be so converted, these are either too complex or are not sufficiently accurate.

In addition to the foregoing, the described arrangement suffers from the disadvantage that the sine and cosine source for 2 and e are, in general, not symmetrically loaded by the measuring divider.

An alternate approach in the tangent measurement technique is to employ a trigonometric divider rather than a linear divider. In a sense, this is equivalent to converting tangent 0 to 0 within the divider, rather than by mechanisms between the divider arm and the output indicator.

Thus, if a single divider winding is tapped at 44 points, spaced or otherwise weighted as a function of tangent rather than linearly, to yield the ratios: tan 1, tan 2, tan 3 tan 45, then the divider setting at null can be directly calibrated as the angle. For angles greater than the practical range, say 46, appropriate switching is necessary whereby the cotangent function would be employed.

Assuming that the required resolution, e.g., to say six places (l0 can be secured at each tap, as by a trimming arrangement, so that a sufficiently accurate ratio representing each 1 point were obtained, there is still the limitation not found in the use of linear decades, viz., that the angular resolution is limited by interpolation problems. Typical requirements call for angular resolution to one thousandth of a degree. Without interpolation, i.e., with a single divider, 45,000 taps would be required in implementing the above approach; this is not, in general, a feasible solution.

Although interpolation between taps, e.g., by cascading further dividers in the manner of cascaded decade dividers, suggests itself as a means of increasing angle resolution, the relation of 0 to tan 0 in the 1 [or any other] increments is not linear. Hence the use of linear interpolation introduces an error which manifests itself in a difference between input 0 and indicated output angle or. notwithstanding a null condition is obtained. Compensation while possible is complicated in that over each interpolation interval a different interpolation error correction is required; the linear interpolation arrangement is thus also burdened with significant disadvantages. In addition, the asymmetrical loading problem also remains.

An alternate technique employs the relationship 00:49 when:

R cos 6 sin a=R sin 0 cos on (B Referring to FIGURE 3A, a first computing divider 30 is shown energized by voltage e :R sin 0 and a second computing divider 31 by e =R cos B, with the dividers having a common connection illustratively connected to ground. The dividers include respective interpolators 40a, 40b.

The movable contacts of dividers 30, 31 and their interpolators are jointly adjusted to set in trial angle a; brushes 30a, 30b, 30c of divider 30, 40a select a ratio representing cosine a while 31a, 31b, 310, of divider 31, 40b are positioned to select sine u. The angle a is the angle read out of the bridge as equal to 0 when proper conditions [e.g., a null] are achieved.

Since divider 30 is energized by R sin 0 and the brushes 30a1-30c thereof are set at cosine a, then the voltage 0 between 300 and ground is equal to R sin 9 cos a [provided there is no significant loading].

Similarly, the voltage e at brush 31c is R cos 9 sin on. These voltages are applied to a meter V. If a is varied by adjusting the divider brushes until e equals e e.g., when these voltages yield a null on meter V, then (1:9 and the setting of the divider brushes represents 9- Hence, and assuming infinite resolution and no errors, 9 has been computed.

It may be noted that the fixed taps of dividers 30 and 31 are trigonometrically distributed or have outputs modified to yield trigonometric functions. The dividers are preferably of identical static construction. Their brushes, however, are located and driven in complementary relationship whereby one represents the sine and the other the cosine function. It may also be noted that the sources 6 and 6 are symmetrically loaded at the null condition. However, two dividers are required in contrast with the single divider used in the tangent technique.

In actual practice dividers 30 and 31 have limited resolution. Additional interpolation for usable resolution is therefore required. In the arrangement shown in FIG- URE 3A the two brushes 31a and 31b selected the values sine 0: and sine a bounding the increment containing the particular value of 9 while interpolation between these values is provided by the interpolating circuit 401). Similarly, interpolator 400, connected to brushes 30a and 30b of divider 30, interpolates between cos a and cos a As noted hereinbefore, the linear interpolation generates an error in a, i.e., in the computed value of 9, provided dividers 40a and 40b are linear rather than trigonometric.

It may be here noted, that the interpolation problem manifests itself in the case of values of 9 which are not equal to the angles represented by the taps on dividers 30 and 31; this follows because the err-or occurs when it is required to interpolate between taps. As angle 9 departs from correspondence with the particular tap value, the error varies. This error may be defined in a number of ways. Since in usage of a practical bridge, the error represents the diiference between the actual value of the input angle 9 and the output or read-out angle on when the bridge is nulled, definition of the error based on this condition is generally referred to herein.

An analysis of the errors generated in the course of interpolation in the circuit of FIGURE 3A indicates that an error function prevails which may be compensated at discrete interpolation points to significantly improve system accuracy. This compensation is schematically indicated in FIGURE 3A by the introduction of appropriate ratio compensation, as signified by e at appropriate points of dividers 40a and 40b. Specific means for ac- I complishing this are illustrated more fully hereinafter.

In contrast with the foregoing interpolation technique, the basic sine-cosine approach of FIGURE 3A may be combined with a ratiometric technique as described below (rather than a linear interpolation with or without compensation) to yield the desired accuracy with a concomitant relative reduction in complexity.

In FIGURE 3B, the voltage 2 representing R cos 9 sin a is combined with the voltage e representing R sin 0 cos a in a differencing type algebraic summing circuit 50 which yields a voltage e When 9 equals (1 the voltage :2 is zero in analogy to the null voltage applied to meter V in FIGURE 3A. As 9 departs from e i.e., Where 9=a +Au, then it can be shown that e varies according to the relationship:

It may be noted that e has the potentiality of indicating the value of Au and thus indicating 9. However R is also an unknown. Hence without more data, Au can not be determined from e,,. If a voltage 0, equal or proportional to R can be obtained, then from (B above, L /c 'a sin dot. A single divider can be employed to calculate the value e /e, whence sin Aot is obtainable without the need for multiple interpolating arrangements such as hereinhefore described. As will be shown below, a simple technique yields a voltage approximately proportional to R Moreover, the error resulting from lack of complete proportionality, when combined with the error associated with the further approximation that dam sin Au, yields a net error which is less than either of its components and which is, moreover, readily compensated. It is to be understood however, that apart from its amenability to sim ple compensation, the generation of the voltage, 6,, as shown below, which is independent of tap position, permits a ratiometric technique as a method of interpolation whereby significant advantages accrue.

As seen in FIGURE 3B, voltages 2 and 2 appear at brushes 30a and 31a respectively, of the dividers 30, 31 and these brushes are each displaced a distance corresponding to a given increment D from the associated brushes 30a and 31a. Hence, the position of brush 300 represents cos (u -l-D) while 31a represents sin (cq-l-D). Relative to ground, the voltages and 42 on brushes 30a, 31a are:

If these two voltages are subtracted in a differencing type algebraic summing circuit such as 51, a voltage, e =e e is obtained which voltage is (where 0:0q-l-Aa as before):

If now e and e,, are subtracted, as in another differencing circuit 52, a voltage (e e,,)=e is obtained, and from 4) and 1),

Analysis will show that e is approximately equal to R -D where D is D modified by a constant to account for the dimensional difference between sine and the angle. Since D is a constant, then e is approximately proportional to R and substantially independent of 11 Hence 9) It can be seen that in this approach sin Act may be derived by a ratiometric technique which is used for computing e /e the interpolation described in connection with FIGURE 3A is thereby avoided. Moerover, it is found that when the further approximation is made that:

then the error resulting from the combined approximatations is less than either component error and takes on a character which is readily compensated to yield an ultimate computation of high accuracy. This result can be seen by an analysis of the error resulting when the ratio e,,/e is rigorously computed as 'by divider 53. Thus,

sin Aa 11) The factor (1 represents the computed or measured value of the ratio. If q is defined as the ratio AOL/D, see (B above, and if values of AOL are assigned to (B above, it will be found that q is different from g by the functions such as f f plotted in IIGURE 3C.

The function h is for D=10, while f represents the increment D=5. It can be shown that peak values of f and f are each about A the peak amplitude of the corresponding functions associated with the straightforward interpolation.

The error functions f f are independent of tap setting (1 and thus admit of a particularly simple compensation technique as will be described below. Hence the required net system accuracy is readily achieved. It should of course be borne in mind that the ratio e /e is actually computed rigorously whereby the above described error is generated. The error for discrete values of within the range D, may be compensated by ratio compensation effected, for example, by injecting into the system, e.g., at divider 53, a voltage e for each of the selected values of 0 within the interpolated increment. For example and with reference to FIGURE 3C, a set of voltages corresponding to the amplitudes of f or f at the values of 0 equal to .lD, .2D, .3D D, can 'be supplied such that the error is reduced to near Zero (e.g., two seconds or less) for these values of 0. Circuit arrangements providing this are described hereinafter. These arrangements are characterized by significant simplicity. For example, they eliminate the need for a divider or alternatively permit a 50% reduction in the number of taps required. Typical savings in costs of or may thereby be realized.

A detailed embodiment of the circuit of FIGURE 3B is illustrated in FIGURE 3D. The bridge system shown therein includes synchro-resolver converter and isolator CV and a resolver type bridge RB energized from the converter. With synchro data supplied at input terminals S S and S and with mode selector SW in the synchro position, then the input synchro data is converted to resolver form and coupled to the bridge RB Adjustment of the bridge via bridge controls BC to achieve a null on meter V, results in the computation of input angle 0 which is illustrated on read-out indicator I With resolver data applied at input terminals RS RS inclusive and with the mode selector SW in the resolver mode, converter CV supplies coupling with isolation, of the resolver data to the bridge, The bridge is adjusted as previously described whereby the resolver angle 0 is computed.

The converter CV comprises transformers T T and T having input windings W W and W respectively. These windings are adapted for connection to the synchro and resolver terminals via switch sections 34e, 34f, 34g and 34h operated by the mode selector SW The switches are shown in the synchro position whereby the input windings are series-connected in delta configuration to the synchro terminals S S and S The transformers T and T each include secondary windings tapped to form two sections W W in the case of T and W and W in the case of T Windings W and W are each shunted by a respective tapped winding W and W which in effect constitute dividers across a part of the secondary for providing the requisite resolution (on separate cores). The tap on W is connected to fixed contacts of switch sections 34c and 34d while the tap on W is connected to switch 34d and additional switch section 34b. These sections along with an additional section 34a are also operated by the mode selector SW Transformer T is provided with a single secondary W which has one end connected to a fixed tap of 34b and the other end to 34a. By tracing the circuit wtih the switches in the synchro mode as shown, it may be noted that the secondary windings of T and T are connected in series-opposition and applied to a divider 41 via reversing switch SW This circuit converts the synchro data to the cosine resolver voltage.

The sine function voltage is developed across winding W of transformer T and in the switch position shown, this voltage is applied to a divider 42 via reversing switch SW The latter and reversing switch SW are switched as required via an actuator 39 which is driven from the bridge control B0 The latter is adjusted in accordance with the 10 increments of bridge angle.

In summary, the above described circuit, when in the illustrated condition, converts the synchro voltages at S S and S to equivalent resolver voltage with the cosine component being applied to divider 41 and the sine component applied to similar divider 42.

With the mode selector switch in the resolver mode it will be noted that the following circuit conditions obtain. Winding W is connected to resolver terminals RS and RS Winding W is connected to resolver terminals RS and RS Secondary Winding W143, is disconnected. The secondary of T which carries the cosine voltage is connected via the reversing switch SW to divider 41. The sine voltage appearing across the secondary of T is applied to divider 42 via SW Hence in the resolver mode the converter is connected to provide coupling with isolation; the switching together with transformers T and T provide these functions.

Dividers 41 and 42 are preferably of identical construction. Each illustratively comprises an auto transformer divider W having 10 taps 41a which are connected to corresponding output taps 41b. The latter are adapted to be selectively contacted by brushes 51a and 51b which are spaced to contact adjacent fixed taps. The brushes 51a and 51b are driven in complementary fashion with respect to corresponding brushes 52a and 52b of divider 42. Thus, the sine and cosine of the selected angle or. are simultaneously derived with identical dividers. The 0 ratio tap of each divider is connected to the reference potential point which is illustratively ground.

Selected taps of each divider 41, 42 have their ratios modified for attaining the requisite resolution. This trimrning may be accomplished in any of a variety of ways including the trimming arrangement used in transformers T T By the appropriate arrangement the required resolution in the trigonometric ratio at each tap is achieved. For example, the voltage added by way of winding 41e can add three additional significant figures to the three-digit ratio attained by the tapped position to yield six-digit resolution at each 10 tap. Generally, no resolution increase or trimming will be required for the sine 0, sine 30, and sine taps. The coarse value of the trigonomometric function at each 10 tap is illustratively attained by varying the number of turns of winding 41 between tap points.

In analogy to the general arrangement illustrated in FIGURE 3B, it may be seen that the voltage at brush 51b is a function of cos 0 sin a while the voltage at 52b varies in accordance with sin 0 cos a The difference between these two voltages is obtained in the illustrated embodiment by connecting the brushes to a winding 53b of a transformer T Hence the voltage across this wind ing constitutes voltage e of FIGURE 3B.

In like manner, the voltages representing cos 6 sin (oq-l-D) and sin 0 cos (oq-l-D) are supplied from brushes 51a and 52a respectively to a winding 53a of a transformer T such that the difference voltage 2 appears across this winding.

The voltages e and e after suitable transformation provided by the associated transformer are combined in accordance with their difference to obtain voltage e Accordingly, secondary winding 54a of T is connected in series-opposition with winding 54b of T The resultant net voltage e is combined with the transformed voltage e to form the ratio e /e In the illustrated embodiment this is accomplished by connecting the windings 54a and 54b across the divider 43, the junction of the two windings being grounded. Divider 43 is preferably an auto transformer divider which is linearly tapped at 1 intervals, as at 43a. The voltages appearing at the taps are modified however, by the injection of respective special compensating voltages e 92, etc. For the S-type error function illustrated in FIGURE 3C, these voltages are relatively proportioned in accordance with that function and hence no corrective voltage is required at the .5D and end taps. This proportioning of the injected compensating voltages has the eifect of correcting ratios thereby reducing the error at the points .1D, .2D, 3D .9D to a very low value and permitting further linear interpolation without introduction of errors of consequence.

The compensated voltages are applied to output taps 43b of the divider 43. Selectively connected to these taps is an interpolating divider comprising three cascaded sec- 

